Tree diagrams are one of the most reliable ways to pick up marks in a GCSE probability question — if you know the method, you can work through them calmly even when the scenario sounds complicated. They appear on all three major boards: AQA 8300 (usually Paper 2 or 3), Edexcel 1MA1 (typically a 4–6 mark question on the calculator paper), and OCR J560. The key is knowing when to use them and how to handle the second branch when events depend on each other.
What You Need to Know First
- Probability is always between 0 and 1
- Probabilities on branches coming from the same point must add up to 1
- To find the probability of two events both happening, multiply along the branches
- To find the probability of two outcomes that both satisfy a condition, add those products
- The difference between independent events (first outcome doesn't affect the second) and dependent events (it does)
Independent Events: Tree Diagrams Without Replacement Concerns
When events are independent, the probabilities on the second set of branches stay the same regardless of what happened on the first branch.
Worked Example 1
A bag contains 3 red counters and 2 blue counters. A counter is picked at random, noted, and put back before a second counter is picked. Find the probability that both counters are the same colour.
Step 1 — Set up the first branches:
- P(Red) = 3/5
- P(Blue) = 2/5
Step 2 — Set up the second branches. Because the counter is replaced, the probabilities are identical:
- After Red: P(Red) = 3/5, P(Blue) = 2/5
- After Blue: P(Red) = 3/5, P(Blue) = 2/5
Step 3 — Multiply along each branch for the four outcomes:
- P(Red, Red) = 3/5 × 3/5 = 9/25
- P(Red, Blue) = 3/5 × 2/5 = 6/25
- P(Blue, Red) = 2/5 × 3/5 = 6/25
- P(Blue, Blue) = 2/5 × 2/5 = 4/25
Step 4 — Add the outcomes that match "same colour": P(same colour) = 9/25 + 4/25 = 13/25
Quick check: all four outcomes should add to 1. 9 + 6 + 6 + 4 = 25. ✓
Dependent Events: Conditional Probability Without Replacement
This is where many students lose marks. When a counter is not replaced, the second set of branches changes — the total number of counters is now one fewer, so every probability on the right-hand branches is different.
Conditional probability means the probability of an event given that something else has already happened. In tree diagram questions, this is built in automatically — you just need to adjust the numbers correctly.
Worked Example 2
Using the same bag (3 red, 2 blue), this time the first counter is not replaced. Find the probability that both counters are red.
Step 1 — First branches remain the same:
- P(Red) = 3/5
- P(Blue) = 2/5
Step 2 — Second branches NOW depend on what happened first. There are now only 4 counters in the bag.
If the first counter was Red (3 red remaining → now 2 red, 2 blue left):
- P(Red | first was Red) = 2/4 = 1/2
- P(Blue | first was Red) = 2/4 = 1/2
If the first counter was Blue (3 red, 1 blue left):
- P(Red | first was Blue) = 3/4
- P(Blue | first was Blue) = 1/4
Step 3 — Multiply along the branch for "both red": P(Red, Red) = 3/5 × 2/4 = 6/20 = 3/10
Step 4 — If the question asked for the probability of one red and one blue (in any order):
- P(Red, Blue) = 3/5 × 2/4 = 6/20 = 3/10
- P(Blue, Red) = 2/5 × 3/4 = 6/20 = 3/10
- P(one of each) = 3/10 + 3/10 = 3/5
Again, check all four outcomes sum to 1: P(RR) + P(RB) + P(BR) + P(BB) = 6/20 + 6/20 + 6/20 + 2/20 = 20/20 = 1 ✓
Common Mistakes Students Make
- Keeping the denominator the same on the second branch when there is no replacement. This is the single most common error. If you pick without replacement, the total drops by 1 after the first pick. Always ask yourself: "Does the first pick change what's left?"
- Adding instead of multiplying along branches. You multiply probabilities along a branch (AND), and add the results of separate branches (OR). Mixing these up is an instant mark loss.
- Forgetting to consider all valid outcomes. Questions like "at least one red" are easiest solved by finding P(no red) and subtracting from 1 — but students sometimes miss a branch combination when listing outcomes directly.
- Not checking that branches sum to 1. If your second-branch probabilities don't add to 1, something is wrong. Catching this before you move on saves you from compounding the error.
- Writing unsimplified fractions when the question asks for an exact answer. You won't lose a mark for leaving an answer as 6/20, but simplifying to 3/10 shows confident handling of fractions and avoids any doubt.
Exam Tips for Tree Diagrams
- Draw the tree fully before answering any part of the question. Rushing to calculate the first answer before the diagram is complete leads to errors on later parts. Two minutes drawing the tree neatly is always worth it.
- Label every branch with its probability and every endpoint with the combined outcome (e.g. "RR", "RB"). This makes it much easier to identify the right paths later, especially in multi-part questions.
- For "at least one" questions, use the complement. P(at least one red) = 1 − P(no red) is almost always quicker than adding three separate outcomes. Examiners award method marks for this approach.
- AQA and Edexcel both award marks for a correctly structured tree diagram even if probabilities are wrong. If you draw the tree correctly but make an arithmetic error, you will still earn the method marks — so show your structure clearly.
- State your working as a multiplication. Write something like P(RR) = 3/5 × 2/4 = 3/10 on the diagram or in your working. Examiners follow your reasoning, and a single slip is less damaging if your method is visible.
- If the question says "given that," this is always a conditional probability question. Identify what you're conditioning on, find the restricted sample space, and recalculate — or use the tree diagram you've already drawn to pick out only the relevant branches.
Practise Now
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